Two bodies have their moments of inertia `I` and `2I`, respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular velocity will be in the ratio

Options

A. `2:1`
B. `1:2`
C. `sqrt(2):1`
D. `1:sqrt(2)`

Explanation

To solve the problem, we need to find the ratio of the angular velocities of two bodies with different moments of inertia but equal kinetic energies of rotation.

### Step-by-Step Solution:

1. **Identify the given data**:
- Moment of inertia of the first body, \( I_1 = I \)
- Moment of inertia of the second body, \( I_2 = 2I \)
- Kinetic energies of both bodies are equal.

2. **Write the formula for kinetic energy of rotation**:
The kinetic energy \( K \) of a rotating body is given by the formula:
\[ K = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

3. **Set up the equation for kinetic energies**:
For body 1:
\[ K_1 = \frac{1}{2} I_1 \omega_1^2 = \frac{1}{2} I \omega_1^2 \]
For body 2:
\[ K_2 = \frac{1}{2} I_2 \omega_2^2 = \frac{1}{2} (2I) \omega_2^2 = I \omega_2^2 \]
Since the kinetic energies are equal, we can set \( K_1 = K_2 \):
\[ \frac{1}{2} I \omega_1^2 = I \omega_2^2 \]

4. **Cancel out common terms**:
We can cancel \( I \) from both sides (assuming \( I \neq 0 \)):
\[ \frac{1}{2} \omega_1^2 = \omega_2^2 \]

5. **Rearrange the equation**:
Multiply both sides by 2:
\[ \omega_1^2 = 2 \omega_2^2 \]

6. **Take the square root**:
Taking the square root of both sides gives:
\[ \frac{\omega_1}{\omega_2} = \sqrt{2} \]

7. **Express the ratio**:
Therefore, the ratio of angular velocities is:
\[ \omega_1 : \omega_2 = \sqrt{2} : 1 \]

### Final Answer:
The ratio of angular velocities \( \omega_1 : \omega_2 \) is \( \sqrt{2} : 1 \).